The MU Problem

Take an initial theorem string of two characters: MI, in an alphabet of three symbols: M, I, U. The MU puzzle asks whether MU can be derived from MI using four rules:

1. If a string ends in I, append U. MxI → MxIU
2. Double everything after M. Mx → Mxx
3. Replace III with U. MxIIIx → MxUx
4. Remove UU. MxUUx → Mxx

At first glance, MU feels attainable. You may try:

MI → MII → MIIII → MUI → MUII → MUIIII...

The path appears open. It is not.

You can never derive MU from MI. Not because of a missing trick, but because of a preserved invariant: the number of I symbols remains constrained in a way that never reaches exactly zero under these transformations. The system permits movement, but not every destination. Its syntax authorizes derivation while quietly forbidding specific outcomes.

This is where the puzzle stops being a puzzle and starts becoming a philosophy of formal systems. Rules do not merely generate expressions; they generate horizons. The legal operations of a system define not only what can be built, but what can never be reached.

The same principle reverberates beyond toy strings:

• why certain arithmetic truths are unprovable in a given formal system, • why some compiler optimizations are impossible in the general case, • why local rewrite rules can imply global impossibilities.

We like to imagine barriers as external, accidental, and temporary. Formal systems remind us that some boundaries are internal, structural, and absolute.

The MU problem is fascinating precisely because it weaponizes simplicity. Four rules. Three symbols. One impossible target. And in that small machine we encounter a hard lesson: effort is not equivalent to reachability.

Sometimes failure is not a deficiency of will. Sometimes it is a property of the system.